%\section{Results and validation}
%\label{sec:risk.results}

\section{Perversity and sidedness}
\label{sec:risk.results.sideness}
We first report on our finding that both one-sided and two-sided
behavior changes can lead to perverse outcomes (less vaccination is
more effective) across a wide range of contact networks.  One-sided
behavior change leads to perverse outcomes at low levels of
intervention, in which the epidemic severity increases with $\pv$, up
to a point, as shown in Figure \ref{fig:scalefree.pt},
\ref{fig:scalefree.ps}, and \ref{fig:scalefree.pb}. Two-sided behavior
change leads to perverse outcomes at high levels of intervention, in
which the epidemic severity starts increasing beyond a threshold value
of $\pv$.  We mathematically establish the phenomena of perversity and
non-monotonicity for graphs generated according to the Erd\"os-Renyi
model \cite{newman:netstructure03}, denoted by $G(n,\pt)$, in which
each edge between a pair of nodes is chosen independently with
probability $\pt$. We prove rigorously that there exist $\pt$, $\pb$,
and $\pf$, such that (i) in the one-sided model, it almost surely
holds that the epidemic severity is $o(n)$ for both $\pv = 0$ and $\pv
= 1$, yet $\Theta(n)$ for some $\pv$ in $\rb{0,1}$; (ii) in the
two-sided model, the epidemic severity is $\Theta(n)$ for both $\pv =
0$ and $\pv = 1$, yet $o(n)$ for some $\pv$ in $\rb{0,1}$.  This
implies that there is a choice of parameters (which turns out to be be
quite broad), such that as the vaccinated fraction $\pv$ is varied,
the epidemic severity shows a non-monotone behavior.

\begin{figure}[htbp]
\begin{center}
\includegraphics[width=2.5in]{figures/pa_1end_035_025_thick.jpg}
\includegraphics[width=2.5in]{figures/pa_2end_035_025_thick.jpg}
\caption{Epidemic severity with different boosted transmissivities in one-sided (left) and two-sided (right) risk behavior models. $x$-axis is the percentage of nodes taking interventions, and $y$-axis is the expected percentage of nodes getting infected. $\pt=0.25$ and $\ps=0.35$.}
\label{fig:scalefree.pt}
\end{center}
\end{figure}

\begin{figure}[htbp]
\begin{center}
\includegraphics[width=2.5in]{figures/pa_1end_ps035_thick.jpg}
\includegraphics[width=2.5in]{figures/pa_2end_ps035_thick.jpg}
\caption{Epidemic severity with different transmissivities in one-sided (left) and two-sided (right) risk behavior models. $x$-axis is the percentage of nodes taking interventions, and $y$-axis is the expected percentage of nodes getting infected. $\ps=0.35$, and $\pb=2\pt$.}
\label{fig:scalefree.ps}
\end{center}
\end{figure}

\begin{figure}[htbp]
\begin{center}
\includegraphics[width=2.5in]{figures/pa_1end_pb05_thick.jpg}
\includegraphics[width=2.5in]{figures/pa_2end_pb05_thick.jpg}
\caption{Epidemic severity with different intervention success probabilities in one-sided (left) and two-sided (right) risk behavior models. $x$-axis is the percentage of nodes taking interventions, and $y$-axis is the expected percentage of nodes getting infected. $\pt=0.25$, and $\pb=0.5$.}
\label{fig:scalefree.pb}
\end{center}
\end{figure}

\begin{theorem}
\label{thm:risk.gnp}
For the Erd\"{o}s-R\'{e}nyi random graph model $G(n,\pt)$, there exist
$\pt$, $\pb$, and $\pf$, such that (i) in the one-sided model, it
almost surely holds that the epidemic severity is $o(n)$ for both $\pv
= 0$ and $\pv = 1$, yet $\Theta(n)$ for some $\pv$ in $\rb{0,1}$; (ii)
in the two-sided model, the epidemic severity is $\Theta(n)$ for both
$\pv = 0$ and $\pv = 1$, yet $o(n)$ for some $\pv$ in $\rb{0,1}$.
\end{theorem}

We give a brief sketch of our proof, which is based on recent results
of S\"oderberg~\cite{soderberg+inhomo02} and Bollob\'{a}s et
al~\cite{bollobas+inhomo} on heterogeneous random graphs.  We refer
the reader to supplementary information for details.  Consider the
model of heterogeneous random graphs denoted by
$\mathcal{G}(N,K,\textbf{r},\textbf{c})$, where (i) $K$ is a positive
integer, (ii) $\textbf{r}=\{r_1,\ldots,r_K\}$ is a probability vector,
(iii) $\textbf{c}=(c_{ij})$ is a $K\times K$ matrix, (iv) each node
$j=1,\ldots,N$, is assigned a type $i\in\{1,\ldots,K\}$ with
probability $r_i$, and (v) each pair of nodes $i, j$ are connected by
an edge with probability
$p(i,j)=c_{ij}/N$. S\"{o}derberg~\cite{soderberg+inhomo02} and
Bollob\'{a}s et al.~\cite{bollobas+inhomo} established the following:
(i) if the eigenvalues of the matrix $\{c_{ij}r_j\}$ are all less than
1, it is sub-critical (i.e., has no giant component), and (ii) if some
eigenvalue is larger than 1, it is super-critical (i.e., has a giant
component) with asymptotically $r_i(1-f_i)N$ nodes of type $i$, where
$f_i$ satisfies the coupled set of equations: $f_i = exp\left(\sum_j
c_{ij}r_j(f_j-1)\right)$.  We show that if the contact network is
generated by the Erdos-Renyi model $G(n,c/n)$, then the disease
transmission process produces a heterogeneous random graph with the
eigenvalue characteristic equation given by
\[
- \lambda(\lambda^2 - (c (1 - \pv) + \pb c \pv \pf) \lambda +
  c^2(1-\pv)\pb\pv\pf - c^2 (1 - \pv) \pv \pf) = 0.
\]
We show the existence of parameters $\pv$, $\pf$, $\pb$, and $c$
such that the absolute value of every eigenvalue is smaller than 1.

We find the phenomenon of perversity exists in a broad class of
graphs, and in order to formally prove its widespread occurrence, we
consider {\em locally finite graphs}, which have been widely studied
in percolation theory (e.g., see~\cite{bollobas:percolationBook}).
Locally finite graphs include infinite graphs in which each node has
bounded degree.  Using techniques from percolation theory, we prove
that in every locally finite graph $G$, there exist $\pt$, $\pb$, and
$\pf$, such that: (i) the epidemic severity is finite for both $\pv =
0$ and $\pv = 1$, yet infinite for some $\pv$ in $\rb{0,1}$ in the
one-sided model; (ii) the epidemic severity is infinite for both $\pv
= 0$ and $\pv = 1$, yet finite for some $\pv$ in $\rb{0,1}$ in the
two-sided model. This result provides strong evidence of the
universality of the phenomenon.  As such it begs for a natural and
intuitive explanation. Our best structural understanding at this point
is that this is the consequence of two competing tensions -- vaccine
success that serves to contain the spread and risky behavior that,
exacerbated by vaccine failure, serves to boost the contagion.  In the
one-sided situation since it is sufficient for infection spread to
have just the one party in an interaction exhibiting risky behavior we
see perversity manifesting itself at low levels of
vaccination. Whereas, in the two-sided situation since it is necessary
for both the interacting parties to exhibit risky behavior we see
perversity manifesting itself only at high vaccination levels which is
a prerequisite for a non-trivial fraction of parties with failed
vaccines to exist.

\begin{theorem}
\label{thm:risk.perc}
For every locally-finite infinite graph $G$, there exist $\pt$, $\pb$,
and $\pf$, such that: (i) the epidemic severity is finite for both
$\pv = 0$ and $\pv = 1$, yet infinite for some $\pv$ in $\rb{0,1}$ in
the one-sided model; (ii) the epidemic severity is infinite for both
$\pv = 0$ and $\pv = 1$, yet finite for some $\pv$ in $\rb{0,1}$ in
the two-sided model.
\end{theorem}
The phenomenon of non-monotonicity and its dependence on sidedness
that we have identified occurs across a wide range of network models.

\subsection{Proof of Theorem \ref{thm:risk.gnp}}
\input{5_risk/gnp}

\section{Randomized vs. targeted vaccinations}
\label{sec:risk.results.target-random}
We next report on our finding that targeted vaccination can be
strictly worse than random vaccination for some level of vaccine
coverage, and this phenomenon occurs both for one-sided as well as
two-sided behavior change (as shown in Figure
\ref{fig:scalefree.target-random}).  In the literature it has been
observed that targeting highly connected individuals for vaccination
lead to better outcomes as opposed to random
coverage~\cite{wang+xhcw:scalefree09,dezso+virus02,berger05}.  Our
finding adds nuance to the existing results when risky behavior is
taken into account.  This counterintuitive phenomenon can also be
explained by the tug of war between successful vaccination and risky
behavior. If the effect of risky behavior is dominant then one would
expect that targeted vaccination ends up being worse than random
coverage since it is the targeted high-degree individuals that are the
most responsible for creating additional contagion. And, in fact the
evidence supports this explanation in that we see targeted coverage
being inferior to random coverage at low levels of vaccination in the
one-sided case but at high levels in the two-sided case.

\begin{figure}[htbp]
\begin{center}
\includegraphics[width=2.5in]{figures/pa_cmp_1end_035_025_thick.jpg}
\includegraphics[width=2.5in]{figures/pa_cmp_2end_035_025_thick.jpg}
\caption{Epidemic severity comparison of random and targeted intervention strategies in one-sided (left) and two-sided (right) risk behavior models. $x$-axis is the percentage of nodes taking interventions, and $y$-axis is the ratio of the epidemic severity in targeted intervention strategy and the epidemic severity in random intervention strategy. $\pt=0.25$, and $\ps=0.35$.}
\label{fig:scalefree.target-random}
\end{center}
\end{figure}


%%%%%%%%%%%%%%%%%%%%%
% simulation section
\input{5_risk/simulation}
